Rigidity Theorems for Diameter Estimates of Compact Manifold with Boundary

Abstract

Let (N,g) be an n-dimensional complete Riemannian manifold with nonempty boundary ∂N. Assume that the Ricci curvature of (N,g) has a negative lower bound Ric≥−(n−1)c2 for some c>0, and the mean curvature of the boundary ∂N satisfies H≥(n−1)c0>(n−1)c for some c0>c>0. Then a known result (cf. [12]) says that supx∈Nd(x,∂N)≤1ccoth−1c0c⁠. In this paper, we prove the rigidity result that if N is compact, then the equality holds if and only if (N,g) is isometric to the geodesic ball of radius 1ccoth−1c0c in an n-dimensional hyperbolic space Hn(−c2) of constant sectional curvature −c2. Moreover, we prove an analogous result for manifold with m-Bakry–Émery Ricci curvature bounded below by a negative constant.